Question

Write the negation of the following statements:
(i). P: For every positive real number x the number $x–1$ is also positive.
(ii). Q: All cats scratch
(iii). R: For every real number x, either $x>1$ or $x<1$
(iv). S: There exists a number x such that $0<x<1$

Answer 1

Hint: In order to find the negation of each of the given statements in the question, we start by understanding the meaning of negation. A negation basically gives us the opposite of the given statement, it contradicts the given. Using this we compute each of the given options.

Complete step-by-step answer:
Given Data,
Negation
By negation we basically intend to contradict the given statement.
“The sun rises in the east.”
The contradiction or negation to this statement is “The sun does not rise in the east”.
Negation of a statement does not have to be true it just has to contradict the given statement.
There can be one or more negations to a given statement.
The negations of the given statements are as follows:
i). P: For every positive real number x the number $x–1$ is also positive.
So we basically try to say there is a real number which does not satisfy the given constraint.
The negation of the statement P is as follows:
There exists a positive real number x such that $x–1$ is not positive.
ii). Q: All cats scratch
We try to contradict that all cats scratch
The negation of the statement Q is as follows:
There exists a cat that does not scratch
iii). R: For every real number x, either $x>1$ or $x<1$
We say that there exists a real number that doesn’t satisfy the above constraints.
The negation of the statement R is as follows:
There exists a real number x such that neither $x>1$ nor $x<1.$
iv). S: There exists a number x such that $0<x<1$
We can say that there is a number that does not satisfy the condition or there is no number that satisfies the condition. Either is correct.
The negation of the statement S is as follows:
There does not exist a number x such that $0<x<1.$

Note: In order to solve this type of problems the key is to know the meaning of a negation to an element and what it exactly does. We just write the exact opposites of the given statements to find the answer.
A negation is a logical operation which basically contradicts the given statement or a function. I.e. if the given statement is true its negation is false and if the given statement is false its negation becomes true.
The negation of a function A is represented as $\sim A$.